The Enigma transformation for each letter can be specified mathematically as a product of permutations. Assuming a three-rotor German Army/Air Force Enigma, let P denote the plugboard transformation, U denote that of the reflector, and L, M, R denote those of the left, middle and right rotors respectively. Then the encryption E can be expressed as
Combining three rotors from a set of five, each of the 3 rotor settings with 26 positions, and the plugboard with ten pairs of letters connected, the military Enigma has 158,962,555,217,826,360,000 different settings (nearly 159 quintillion or about 67 bits).
Note that (5x4x3) x (26^3) x [26! / (6! x 10! x 2^10)] = 158,962,555,217,826,360,000 ≈ 267.1